Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
(for now, focus on metals)
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?
Chapter 3 - 1
Energy and Packing
• __________, ________ packing
Energy
typical neighbor
bond length
typical neighbor
bond energy
• ________, __________ packing
r
Energy
typical neighbor
bond length
r
typical neighbor
bond energy
________, _________packed structures tend to have
lower energies.
Chapter 3 - 2
Materials and Packing
______________ materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers
crystalline SiO2
Adapted from Fig. 3.22(a),
Callister 7e.
__________________ materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
”______________" = Noncrystalline
Si
Oxygen
noncrystalline SiO2
Adapted from Fig. 3.22(b),
Callister 7e.
Chapter 3 - 3
Section 3.3 – Crystal Systems
___________: smallest repetitive volume which
contains the complete lattice pattern of a crystal.
crystal systems
crystal lattices
a, b, and c are the lattice constants
Fig. 3.4, Callister 7e.
Chapter 3 - 4
Section 3.4 – Metallic Crystal Structures
• How can we stack metal atoms to minimize
empty space?
2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures
Chapter 3 - 5
Metallic Crystal Structures
• Tend to be densely packed.
• Reasons for dense packing:
- Typically, only one element is present, so all atomic
radii are the same.
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be small in
order to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Chapter 3 - 6
Simple Cubic Structure (SC)
• Rare due to low packing denisty (only Po has this structure)
• Close-packed directions are ____________________.
• Coordination # =
(# nearest neighbors)
(Courtesy P.M. Anderson)
Chapter 3 - 7
Atomic Packing Factor (APF)
APF =
*assume hard spheres
• APF for a simple cubic structure =
a
APF =
Adapted from Fig. 3.23,
Callister 7e.
Chapter 3 - 8
Body Centered Cubic Structure (BCC)
• Atoms touch each other __________________________.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Cr, W, Fe (α), Tantalum, Molybdenum
• Coordination # =
Adapted from Fig. 3.2,
Callister 7e.
(Courtesy P.M. Anderson)
Chapter 3 - 9
Atomic Packing Factor: BCC
• APF for a body-centered cubic structure =
3a
a
2a
Adapted from
Fig. 3.2(a), Callister 7e.
R
a
Close-packed directions:
length =
atoms
APF =
Chapter 3 - 10
Face Centered Cubic Structure (FCC)
• Atoms touch each other along _____________________.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # =
Adapted from Fig. 3.1, Callister 7e.
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
(Courtesy P.M. Anderson)
Chapter 3 - 11
Atomic Packing Factor: FCC
• APF for a face-centered cubic structure =
maximum achievable APF
Close-packed directions:
length =
2a
Unit cell contains:
=
a
Adapted from
Fig. 3.1(a),
Callister 7e.
atoms
unit cell
APF =
volume
atom
volume
unit cell
Chapter 3 - 12
FCC Stacking Sequence
• _________________... Stacking Sequence
• 2D Projection
B
A sites
B sites
A
B
C
B
C
B
B
C
B
B
C sites
• FCC Unit Cell
A
B
C
Chapter 3 - 13
Hexagonal Close-Packed Structure
(HCP)
• ____________... Stacking Sequence
• 3D Projection
c
a
• 2D Projection
A sites
Top layer
B sites
Middle layer
A sites
Bottom layer
Adapted from Fig. 3.3(a),
Callister 7e.
• Coordination # =
• APF =
• c/a = 1.633
__________________
ex: Cd, Mg, Ti, Zn
Chapter 3 - 14
Theoretical Density, ρ
Density = ρ =
Mass of Atoms in Unit Cell
Total Volume of Unit Cell
ρ =
where
n = ___________________
A = ___________________
VC = ___________________
NA = Avogadroʼs number
= 6.023 x 1023 atoms/mol
Chapter 3 - 15
Theoretical Density, ρ
• Ex:
A=
R=
n=
R
atoms
unit cell
ρ=
a
g
mol
ρtheoretical
ρactual
volume
atoms
unit cell
mol
Chapter 3 - 16
Densities of Material Classes
In general
ρ_______> ρ_______ > ρ_________
30
Why?
20
Metals have...
• less dense packing
• often lighter elements
Polymers have...
ρ (g/cm3 )
• close-packing
10
(metallic bonding)
• often large atomic masses
Ceramics have...
• low packing density
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Metals/
Alloys
5
4
3
2
1
0.5
0.4
0.3
Platinum
Gold, W
Tantalum
Silver, Mo
Cu,Ni
Steels
Tin, Zinc
Titanium
Aluminum
Magnesium
Graphite/
Ceramics/
Semicond
Composites/
fibers
Polymers
B ased on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,
Carbon, & Aramid Fiber-Reinforced
Epoxy composites (values based on
60% volume fraction of aligned fibers
in an epoxy matrix).
Zirconia
Al oxide
Diamond
Si nitride
Glass -soda
Concrete
Silicon
G raphite
PTFE
Silicone
PVC
PET
PC
HDPE, PS
PP, LDPE
Glass fibers
GFRE*
Carbon fibers
CFRE*
A ramid fibers
AFRE *
Wood
Data from Table B1, Callister 7e.
Chapter 3 - 17
Crystals as Building Blocks
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
(Courtesy Martin Deakins,
GE Superabrasives,
Worthington, OH. Used with
permission.)
--turbine blades
Fig. 8.33(c), Callister 7e.
(Fig. 8.33(c) courtesy
of Pratt and Whitney).
• Properties of crystalline materials
often related to crystal structure.
--Ex: Quartz fractures more easily
along some crystal planes than
others.
(Courtesy P.M. Anderson)
Chapter 3 - 18
Polycrystals
• Most engineering materials are polycrystals.
1 mm
• Nb-Hf-W plate with an electron beam weld.
• Each "grain" is a single crystal.
• If grains are randomly oriented,
Anisotropic
Adapted from Fig. K,
color inset pages of
Callister 5e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
Isotropic
overall component properties are not directional.
• Grain sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
Chapter 3 - 19
Single vs Polycrystals
• Single Crystals
E (diagonal) = 273 GPa
Data from Table 3.3,
Callister 7e.
(Source of data is R.W.
Hertzberg, Deformation
and Fracture Mechanics
of Engineering
Materials, 3rd ed., John
Wiley and Sons, 1989.)
-Properties vary with
direction: _______________.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: ___________.
(Epoly iron = 210 GPa)
-If grains are __________,
anisotropic.
E (edge) = 125 GPa
200 µm
Adapted from Fig. 4.14
(b), Callister 7e.
(Fig. 4.14(b) is courtesy
of L.C. Smith and C.
Brady, the National
Bureau of Standards,
Washington, DC [now
the National Institute of
Standards and
Technology,
Gaithersburg, MD].)
Chapter 3 - 20
Section 3.6 – Polymorphism
• Two or more distinct crystal structures for the same
material (allotropy/polymorphism)
iron system
titanium
liquid
α, β-Ti
1538ºC
δ-Fe
BCC
carbon
1394ºC
diamond, graphite
γ-Fe
FCC
912ºC
BCC
α-Fe
Chapter 3 - 21
Crystal Structure Review
•
Unit cell- smallest number of atoms showing atomic
arrangement in a crystal
•
•
•
•
Cubic system: a=b=c, α=β=γ=90o
Structure #atoms/uc atom positions
APF Coord #
a
Simple
1
corners
0.52
6
2R
BCC
2
corners+middle
0.68
8
4R/
√3
FCC
4
corners+face centers 0.74 12
2√2R
Density=(#atoms/uc)(mass/atom)/(volume of unit cell)
•
•
Chapter 3 - 22
Section 3.8 Point Coordinates
z
Point coordinates for unit cell
center are
111
c
a/2, b/2, c/2
000
a
x
y
b
Point coordinates for unit cell
corner are 111
•
z
½½½
2c
•
•
•
b
y
Translation: integer multiple of
lattice constants à identical
position in another unit cell
b
Chapter 3 - 23
Point Coordinates-2
Chapter 3 - 24
Crystallographic Points
Chapter 3 - 25
Crystallographic Directions
Algorithm
z
y
x
1. Read off point coordinates of _____ and
_________in terms of unit cell dimensions a,
b, and c
2.Calculate vector (____________________)
3. Adjust to __________________________
4. Enclose in _____________ brackets, no
commas
_____________________
families of directions _____________
Chapter 3 - 26
Crystallographic Directions
Chapter 3 - 27
HCP Crystallographic Directions
z
Algorithm
a2
-
a3
a1
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvtw]
a
2
ex:
½, ½, -1, 0
-a3
a2
2
Adapted from Fig. 3.8(a), Callister 7e.
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
Chapter 3 - 28
HCP Crystallographic Directions
• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are
related to the direction indices (i.e., u'v'w') as
follows.
z
[ u 'v 'w ' ] → [ uvtw ]
1
(2 u ' - v ')
3
1
=
v
(2 v ' - u ')
3
t = - (u +v )
u=
a2
-
a3
a1
Fig. 3.8(a), Callister 7e.
w = w'
Chapter 3 - 29
Crystallographic Planes
Adapted from Fig. 3.9, Callister 7e.
Chapter 3 - 30
Crystallographic Planes
• Miller Indices: Reciprocals of the (three) axial
intercepts for a plane, cleared of fractions &
common multiples. All parallel planes have
same Miller indices.
• Algorithm
1. Planes cannot pass _______________________
2. Planes must _________ or be ___________ to the
axes
3. Read off ______________ of plane with axes in
terms of a, b, c
4. Take _______________ of intercepts
5. Reduce to ________________values
6. Enclose in _________________, no
commas i.e., ___________
Chapter 3 - 31
Crystallographic Planes
z
example
a
b
c
c
b
a
x
example
a
b
y
z
c
c
a
b
x
Chapter 3 - 32
y
Crystallographic Planes
z
example
a
b
c
c
•
a
•
•
b
y
x
Family of Planes _______
Chapter 3 - 33
Crystallographic Planes (HCP)
• In hexagonal unit cells the same idea is used
z
example
a1
a2
a3
c
a2
a3
a1
Adapted from Fig. 3.8(a), Callister 7e.
Chapter 3 - 34
Crystallographic Planes
•
•
We want to examine the atomic packing of
crystallographic planes
Iron foil can be used as a catalyst. The
atomic packing of the exposed planes is
important.
a) Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these
planes.
Chapter 3 - 35
Planar Density of (100) Iron
Solution: At T < 912°C iron has the BCC structure.
2D repeat unit
(100)
Adapted from Fig. 3.2(c), Callister 7e.
a=
4 3
R
3
Radius of iron R = 0.1241 nm
Chapter 3 - 36
Planar Density of (111) Iron
Solution (cont): (111) plane
1 atom in plane/ unit surface cell
2a
atoms in plane
atoms above plane
atoms below plane
h=
3
a
2
2
⎛ 4 3 ⎞ 16 3 2
area = 2 ah = 3 a = 3 ⎜⎜
R ⎟⎟ =
R
3
⎝ 3
⎠
2
Chapter 3 - 37
SUMMARY
• Atoms may assemble into crystalline or
amorphous structures.
• Common metallic crystal structures are FCC, BCC, and
HCP. Coordination number and atomic packing factor
are the same for both FCC and HCP crystal structures.
• We can predict the density of a material, provided we
know the atomic weight, atomic radius, and crystal
geometry (e.g., FCC, BCC, HCP).
• Crystallographic points, directions and planes are
specified in terms of indexing schemes.
Chapter 3 - 38
SUMMARY
• Materials can be single crystals or polycrystalline.
Material properties generally vary with single crystal
orientation (i.e., they are anisotropic), but are generally
non-directional (i.e., they are isotropic) in polycrystals
with randomly oriented grains.
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
Chapter 3 - 39